Developing a New Mathematical Model for 1:1 Resonant Equatorial Orbits under a J2 Perturbation

Abstract

The goal of this paper is to develop a more accurate model for describing 1:1 resonant Earth equatorial orbits, commonly referred to as geostationary orbit. These orbits are particularly important for communications satellites, weather satellites, and any other satellite that needs to remain at a fixed location over the Earth. Although current methods for calculating these orbits are not wrong, they do not always provide a good indication of the physical motion of satellites under perturbations. For certain perturbed orbits, the solution to the equations of motion describes an osculating orbit that is not a geometrically accurate representation of the actual trajectory. This paper will be looking at the mathematics behind orbits perturbed from secondary terms in the Earth’s gravitational potential. Using the J2 term as a starting point, since it is by far the largest perturbation in the Earth’s potential, we will begin with the basic equations of motion and work through all the necessary steps to highlight the faults with current methods. We will then modify the equations in order to produce a solution that more closely models the true orbit of the satellite, and use these new equations to look at perturbations from higher terms in the Earth’s potential